Type-II Dirac photons

Weak disorders that preserve the crystalline symmetry on average should preserve the Dirac points and their topological surface states.

“Weak disorders that preserve the crystalline symmetry on average should preserve the DPs and their topological surface states.29,47”

• cites [29] Wang, H.-X., Xu, L., Chen, H. Y. & Jiang, J.-H. Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016). sourcePartly supported

  Appendix H (p. 10) says that weak fabrication errors can preserve the Dirac points. Appendix E (pp. 8–10) describes topological boundary states in an inversion-symmetry-broken structure, but does not establish that weak disorder preserves those states. The paper also does not address the claim's specific condition that crystalline symmetry be preserved on average. (LLM, conf=0.88)

  Experimental — accuracy not yet measured
• cites [47] Fu, L. & Kane, C. L. Topology, delocalization via average symmetry and the symplectic Anderson transition. Phys. Rev. Lett. 109, 246605 (2012). sourcePartly supported

  The discussion of WTI and TCI surfaces (page not indexed) says that average translation or mirror symmetry prevents localization even under strong disorder, supporting the claim about robust surface states. It does not establish that disorder preserves the Dirac points themselves; its result concerns delocalization of surface states. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

A Dirac point can be regarded as composed of a pair of Weyl points of opposite Chern numbers.

“A DP can be regarded as composed of a pair of WPs of opposite Chern numbers.”

• cites [29] Wang, H.-X., Xu, L., Chen, H. Y. & Jiang, J.-H. Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016). sourceSupported

The dispersion of the type-II Dirac point in the kx-ky plane is distinctive from existing type-I Dirac points.

“The dispersion of the type-II DP in the k x -k y plane is distinctive from the existing type-I DPs 29,31,32 (see Fig. 5d).”

• cites [29] Wang, H.-X., Xu, L., Chen, H. Y. & Jiang, J.-H. Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016). sourceSupported
• cites [31] Lu, L. et al. Symmetry-protected topological photonic crystal in three dimensions. Nat. Phys 12, 337–340 (2016). sourceCould not verify

  The discussion of Fig. 1a compares a three-dimensional generalized Dirac point with a regular three-dimensional Dirac point, while the Fig. 2 caption describes anisotropic surface-cone dispersion in the kx–ky plane. Neither passage identifies a type-II Dirac point or compares its in-plane dispersion with type-I Dirac points. Searches for “type-II Dirac point” and “tilted Dirac cone” found no passage addressing that distinction. (LLM, conf=0.74)

  Experimental — accuracy not yet measured
• cites [32] Slobozhanyuk, A. et al. Three-dimensional all-dielectric photonic topological insulator. Nat. Photon 11, 130–136 (2017). sourceSupported

When space symmetry is reduced, Weyl points can emerge from Dirac points.

“Thus when the space symmetry is reduced WPs can emerge from DPs. 29”

• cites [29] Wang, H.-X., Xu, L., Chen, H. Y. & Jiang, J.-H. Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016). sourceSupported

A Dirac point with synthetic Kramers double degeneracy and parity-inversion has nontrivial topological properties.

“It has been shown in ref. 29 that a DP with synthetic Kramers double degeneracy and parity-inversion has nontrivial topological properties.”

• cites [29] Wang, H.-X., Xu, L., Chen, H. Y. & Jiang, J.-H. Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016). sourceSupported

The (100) surface states of the tetragonal photonic crystal reveal the Z2 topology of the Dirac points.

“According to the bulk-edge correspondence principle, [2][3][4] the (100) surface states of the tetragonal PhC can reveal the Z 2 topology of the DPs.”

• cites [2] Hasan, M. Z. & Kane, C. L. Topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010). sourcePartly supported

  “A. Strong and weak topological insulators” (p. 10) explains that surface-state patterns reflect bulk Z₂ invariants, supporting the bulk-boundary principle invoked by the citation. The paper does not address the (100) surface of a tetragonal photonic crystal or establish that its surface states reveal the topology of its Dirac points. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites [3] Qi, X.-L. & Zhang, S.-C. Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011). sourceSupported
• cites [4] Vafek, O. & Vishwanath, A. Dirac fermions in solids: from high-T c cuprates and graphene to topological insulators and Weyl semimetals. Ann. Rev. Cond. Matt. Phys 5, 83–112 (2014). sourceSupported

Such Dirac points are monopoles of the SU(2) Berry-phase gauge fields.

“In fact, such DPs are monopoles of the SU(2) Berryphase gauge fields.29”

• cites [29] Wang, H.-X., Xu, L., Chen, H. Y. & Jiang, J.-H. Three-dimensional photonic Dirac points stabilized by point group symmetry. Phys. Rev. B 93, 235155 (2016). sourceSupported

Screw symmetries lead to double degeneracy for all Bloch states on certain planes in the Brillouin zone.

“It is known that screw symmetries lead to double degeneracy for all Bloch states on certain planes in the Brillouin zone (BZ).36,37”

• cites [36] König, A. & Mermin, N. D. Electronic level degeneracy in nonsymmorphic periodic or aperiodic crystals. Phys. Rev. B 56, 13607–13610 (1997). sourceNot checked — no full text of the source
• cites [37] Parameswaran, S. A., Turner, A. M., Arovas, D. P. & Vishwanath, A. Topological order and absence of band insulators at integer filling in non-symmorphic crystals. Nat. Phys. 9, 299–303 (2013). sourcePartly supported

  The Band Theory section (p. 2) says nonsymmorphic symmetry prevents certain bands from being separated, and Fig. 2 (p. 4) shows screw-symmetry-enforced contacts between pairs of bands. The copy does not say that every Bloch state is doubly degenerate across particular Brillouin-zone planes; its discussion of invariant planes in the Fourier-Space Approach (p. 7) concerns a many-body degeneracy argument, not that claim. (LLM, conf=0.86)

  Experimental — accuracy not yet measured

The photonic crystal architecture allows realization of type-II Weyl points using dissipationless all-dielectric materials.

“Our PhC architecture thus allows realization of type-II WPs using dissipationless all-dielectric materials.”

• AssumptionRests on an assumption: The all-dielectric materials used in the proposed architecture are dissipationless or have very low energy dissipation.

The emergent type-II Dirac points carry nontrivial topology.

“The emergent type-II Dirac points carry nontrivial topology and are the mother states of type-II Weyl points.”

The two-fold degeneracy is restored only when qx = 0 or qy = 0, in accordance with the screw symmetries.

“The two-fold degeneracy is restored only when q x = 0 or q y = 0, in accordance with the screw symmetries.”

Both type-I and type-II Dirac points have the same Z2 topology.

“Nonetheless, both type-I and type-II DPs have the same Z 2 topology (see Fig. 2d).”

Accidental degeneracy between bands of opposite parity is protected by the C2 symmetry.

“However, accidental degeneracy between bands of opposite parity is protected by the C 2 symmetry.”

The emergence of Dirac points is robust to the shape and permittivity of the dielectric blocks.

“Fig. 2e shows that the emergence of DPs is quite robust to the shape and permittivity of the dielectric blocks (more examples are given in the Supplementary Materials), since any crossing between bands of different parities on the MA line leads to DPs.”

Both type-II and type-I Weyl points can be derived from these Dirac points when symmetry is reduced.

“Moreover, we show that both type-II and type-I WPs can be derived from these DPs when symmetry is reduced.”

The frequency of a conventional photonic crystal cavity state with woodpile cladding is much more sensitive to the thickness of the slab.

“In contrast, the frequency of a conventional PhC cavity state with woodpile-PhC cladding is much more sensitive to the thickness of the slab 42 (see inset of Fig. 4c and details in Methods), despite the fact that the woodpile PhC has a large complete photonic band gap of δω/ω = 21% while our PhC has no complete photonic band gap.”

• cites [42] John, S. & Yang, S. Electromagnetically Induced Exciton Mobility in a Photonic Band Gap. Phys. Rev. Lett. 99, 046801 (2007). sourceNot checked — no full text of the source

From the unique spectral properties of type-II Dirac points, anomalous refraction of light is derived: two concurrent refraction beams of opposite angles.

“From the unique spectral properties of the type-II DPs, using frequency and wavevector matching, we derive the anomalous refraction of light: there are two concurrent refraction beams of opposite angles (see schematic in Fig. 5g).”

The sign of the photonic total angular momentum is changed when the wavevector is reversed.

“The sign of the photonic TAM is changed when the wavevector is reversed (see Fig. 4b).”

Anomalous refraction with one or two pairs of opposite refraction angles is predicted for type-II Dirac points/Weyl points.

“Anomalous refraction with one or two pairs of opposite refraction angles is predicted for type-II DPs/WPs.”

Screw symmetry is an effective tool for the creation of Dirac points.

“Besides, we unveil screw symmetry, a fundamental type of nonsymmorphic symmetry, as an effective tool for the creation of DPs.”

The topological surface states carry finite total angular momentum.

“The topological surface states carry finite TAM as indicated in Fig. 4b by the winding profile of the Poynting vectors.”

There are six Weyl points identified in Fig. 3b.

“We identify six WPs in Fig. 3b (there are more WPs at higher frequency, explaining the nonzero total Chern number).”

The isofrequency contour near a type-II Dirac point is a hyperbolic curve.

“The isofrequency contour near a type-II DP is a hyperbolic curve (Fig. 5b).”

Concurrent negative and positive refraction can be realized in 3D all-dielectric photonic crystals through type-II Dirac points/Weyl points.

“Here we find, from frequencywavevector conservation, that concurrent negative and positive refraction can also be realized in 3D all-dielectric PhCs through type-II DPs/WPs.”

Zero refraction angle is realized when φ_i is close to π/4, 3π/4, 5π/4, or 7π/4, due to vanishing group velocity in the kx-ky plane.

“Zero refraction angle is realized when ϕ i is close to π 4 ; 3π 4 ; 5π 4 , or 7π 4 , due to vanishing group velocity in the k x -k y plane.”

There are four Dirac points in the first six bands, due to crossing between p- and d-doublets.

“Figure 2d shows that there are four DPs in the first six bands, due to the crossing between the p-and d-doublets.”

The isofrequency contours near a type-I Dirac point are elliptical.

“In contrast, the isofrequency contours near a type-I DP is of elliptical shapes.”

All Bloch states are doubly degenerate on the ky = π plane due to Θ_y^2 = -1.

“Similarly, all Bloch states are doubly degenerate on the k y = π plane due to Θ 2 y ¼ À1.”

The spectrum is nondegenerate for finite qx and qy.

“This spectrum is nondegenerate for finite q x and q y .”

The two degenerate states in any doublet have the same eigenvalue of the C2 rotation.

“Elegantly, the above algebra reveal that the two degenerate states in any doublet have the same eigenvalue of the C 2 rotation.”

The proposed all-dielectric architecture enables robust cavity states at photonic-crystal-air interfaces.

“The proposed all-dielectric architecture enables robust cavity states at photonic-crystal-air interfaces and anomalous refraction, with very low energy dissipation.”

The removal of the two screw symmetries lifts the double degeneracy on the MA line.

“The removal of the two screw symmetries lifts the double degeneracy on the MA line.”

All photonic states on the kx = π plane are doubly degenerate.

“The above equation, as an analog of the Kramers theorem for fermions, guarantees that all photonic states on the k x = π plane are doubly degenerate (see Fig. 2c).”

The crossings between the p and d bands result in Weyl points of Chern number ±1.

“The crossings between the p and d bands results in WPs of Chern number ±1 (see Supplementary Materials for a k • P analysis).”

A gapless surface band traverses the projected photonic band gap.

“Figure 4a shows a gapless surface band traversing the projected photonic band gap.”

The proposed all-dielectric architecture enables anomalous refraction.

“The proposed all-dielectric architecture enables robust cavity states at photonic-crystal-air interfaces and anomalous refraction, with very low energy dissipation.”

The topological surface states can exist on the photonic-crystal-air interface without further cladding, even though such interface does not preserve the screw symmetries.

“The topological surface states here can exist on the PhC-air interface without further cladding, even though such interface does not preserve the screw symmetries.”

Dirac points emerge in pairs with opposite topological charge N_DP at opposite wavevectors.

“Since the total topological charge of photonic bands in the BZ is strictly zero, DPs emerge in pairs with opposite N DP at opposite wavevectors.”

Since the two Dirac points at opposite wavevectors have opposite η, this property can be exploited for selective excitation of type-II Dirac points.

“Since the two DPs at opposite wavevectors have opposite η, the above property can be exploited for selective excitation of type-II DPs.”

Type-II Dirac points are the mother states of type-II Weyl points.

“The emergent type-II Dirac points carry nontrivial topology and are the mother states of type-II Weyl points.”

Any crossing between bands of different parities on the MA line leads to Dirac points.

“Fig. 2e shows that the emergence of DPs is quite robust to the shape and permittivity of the dielectric blocks (more examples are given in the Supplementary Materials), since any crossing between bands of different parities on the MA line leads to DPs.”

The topological surface states form resilient, subwavelength quasi-2D photonic systems.

“The topological surface states thus form resilient, subwavelength quasi-2D photonic systems.”

The gapless surface band is induced by the type-I topological Dirac points.

“Thus the gapless surface band is induced by the type-I topological DPs.”

With two orthogonal screw axes, the product of the two screw rotations is essentially the parity required by the Dirac points.

“The screw symmetries become particularly powerful when there are two orthogonal screw axes, since the product of the two screw rotations is essentially the parity required by the DPs.”

The angle between the crossing lines sets the bounds on the refraction angles near a type-II Dirac point as ±(1/2π - θ_DP).

“This quantity sets the bounds on the refraction angles near a type-II DP as ± 1 2 π À θ DP ð Þ .”

The unconventional optical properties also hold for type-II Weyl points.

“The above unconventional optical properties also holds for type-II WPs.”

The proposed photonic crystals can in principle be fabricated using layer-by-layer methods with current technology for infrared frequencies.

“These PhCs can in principle be fabricated using layer-by-layer methods with the current technology 38,39 for infrared frequencies.”

• cites [38] Lee, J.-H. et al. Layer-by-layer photonic crystal fabricated by low-temperature atomic layer deposition. Appl. Phys. Lett. 90, 151101 (2007). sourceNot checked — no full text of the source
• cites [39] Deubel, M. et al. Direct laser writing of three-dimensional photonic-crystal templates for telecommunications. Nat. Mater. 3, 444–447 (2004). sourceNot checked — no full text of the source

Topology can be a possible tool to suppress inhomogeneous broadening, which is a main obstacle for scalable optical and quantum devices.

“This suggests that topology can be a possible tool to suppress inhomogeneous broadening which is a main obstacle for scalable optical and quantum devices.”

The robustness of the surface photonic bands shows superiority over conventional photonic crystal cavity states.

“The robustness of the surface photonic bands show superiority over conventional PhC cavity states.”